On Khovanov Homology and Related Invariants

Hyperbolic Knots and Torsion in Khovanov Homology

“…The constructions used in Lemmas 3.1 and 3.2 apply to any link homology theory for which Theorem 2.1 is true. Hence, they hold for knot Floer homology [37] and sl(n)-homology [7].…”

“…In [13], Daemi et al prove that any flavor of Heegard-Floer homology applied to the 2-fold branch cover of a ribbon concordance also induces an injection. Caprau et al [7] showed that Theorem 2.1 holds for the sl(n)-homology functor for all n ≥ 2. An axiomatic TQFT approach was used by Kang [16] to prove Theorem 2.1 for a wide class of link homology theories.…”

Split link detection for sl(P)$\mathfrak {sl}(P)$ link homology in characteristic P$P$

“…By [27], there is a ribbon concordance $$C_b$$ from $$K_\#$$ to $$K_b$$. There is an induced map $$$$\begin{equation*} \smash{\overline{\operatorname{KR}}}_N(C_b;\mathbf {F})\colon \smash{\overline{\operatorname{KR}}}_N(K_\#,q;\mathbf {F}) \rightarrow \smash{\overline{\operatorname{KR}}}_N(K_b,q;\mathbf {F}), \end{equation*}$$$$which can be shown to be injective [8, 19] ultimately based on an argument of Zemke for knot Floer homology [44]. By the proof of [40, Proposition 5.7], there are injective maps $$\smash{\overline{\operatorname{KR}}}_N(C_s;\mathbf {F})$$ and $$\smash{\overline{\operatorname{KR}}}_N(C_\ell ;\mathbf {F})$$ (displayed below as dotted) making the diagram commute.…”

“…By [27], there is a ribbon concordance 𝐶 𝑏 from 𝐾 # to 𝐾 𝑏 . There is an induced map KR 𝑁 (𝐶 𝑏 ; 𝐅)∶ KR 𝑁 (𝐾 # , 𝑞; 𝐅) → KR 𝑁 (𝐾 𝑏 , 𝑞; 𝐅), which can be shown to be injective [8,19] ultimately based on an argument of Zemke for knot Floer homology [44]. By the proof of [40,Proposition 5.7], there are injective maps KR 𝑁 (𝐶 𝑠 ; 𝐅) and KR 𝑁 (𝐶 𝓁 ; 𝐅) (displayed below as dotted) making the diagram commute.…”

Nonorientable link cobordisms and torsion order in Floer homologies

“…Ours is one of several recent papers related to ribbon cobordisms. Zemke [33] showed that knot Floer homology obstructs ribbon concordance, a result that prompted a flurry of interesting results in this area, including Levine and Zemke [21], Miller and Zemke [23], Daemi, Lidman, Vela-Vick and Wong [5], Kang [13] and Caprau, González, Lee, Lowrance, Sazdanović and Zhang [4]. Other papers in the area are Sarkar's paper on the ribbon distance [31] and the already-cited paper of Juhász, Miller and Zemke [11], which is the closest paper to ours.…”